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Friday, August 08, 2008

Cool Cats II

One cannot overemphasize how cool it is to think of a two point set as an example of a system of mutually unbiased bases.

Now ordinary categories are not as interesting as toposes, so we would like to extend the comparison of sets and vector spaces to generalised axioms for higher dimensional toposes. But then we had better consider multicategories as well. This is what logos theory aims to do: utilise physically important structures such as MUBs to determine a recursive series of axioms for $n$-dimensional topos like structures. Since ordinary weak $n$ categories should appear as certain algebras, this is clearly a very difficult problem. Unfortunately, working on gravity is like throwing oneself constantly against the brick wall of major problems in mathematics, so a poor physicist just has to accept this fate.

The dimension of the matrix operators that we study is the dimension $n$ of the MUB problem, which states that there are always $n + 1$ MUBs for prime power dimension. This $n$ also represents the dimensionality of the logos structure. Hence sets, or vector spaces, are two dimensional beasts, partly because the subobject classifier is specified by two elements, but more importantly because the categorical structure has two levels of arrow and the classifying square is planar.

Hence the M theorist's obsession with three object groupoids and triangles and cubes and other trinities. Most 20th century mathematics is set theoretic, meaning that it only deals with one dimensional logoses. For example, a group is usually treated as a set with extra structure (multiplication, unit, associativity). But as a category a group only has one object. It is often more natural to use groupoids, such as the groupoid of directed loops in a space, with concatenation for composition. All the loops from a given point to itself form a group anyway, so in a two point space there would be two groups sitting inside the loop groupoid.

4 Comments:

I think I'll have to invite you over here to explain all these things to Bill, Ross, me and the others cause it's far beyond the things I can currently grasp. If you'e interested I'll see if I can dig out some money from some source for funding such a trip.

Oh my god, Bob, are you serious? Of course I'm interested! I can't think of anywhere else I'd rather visit, since it's one of the great centres of category theory! And I would love to have some time to get back into the pure categorical side of things. My only problem is that I'm far too poor to do without an income, but at present I have the advantage of being homeless, so I don't have to worry about paying rent while I'm away.

The part time teaching at UC is divided up between the postgrads. They don't pay well, and it isn't enough money to live off. Then I am not qualified to teach at a school, and this would take up too much of my time anyway.